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2 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 2 Xajun Xng and ngl Yuan Nrthwestern Plytechncal Unversty, X an, Chna 1. Intrductn Quanttatve feedback thery (hereafter referred as QFT), develped by Isaac Hrwtz (Hrwtz, 1963; Hrwtz and Sd, 1972), s a frequency dman technque utlzng the Nchls chart n rder t acheve a desred rbust desgn ver a specfed regn f plant uncertanty. esred tme-dman respnses are transfrmed nt frequency dman tlerances, whch lead t bunds (r cnstrants) n the lp transmssn functn. The desgn prcess s hghly transparent, allwng a desgner t see what trade-ffs are necessary t acheve a desred perfrmance level. QFT s als a unfed thery that emphaszes the use f feedback fr achevng the desred system perfrmance tlerances despte plant uncertanty and plant dsturbances. QFT quanttatvely frmulates these tw factrs n the frm f (a) the set { T } f acceptable cmmand r trackng nput-utput relatnshps and the set { T } f acceptable dsturbance nput-utput relatnshps, and (b) a set { P} f pssble plants whch nclude the uncertantes. The bjectve s t guarantee that the cntrl rat T Y / s a member f and T Y / s a member f, fr all plants P whch are cntaned n. QFT has been develped fr cntrl systems whch are bth lnear and nnlnear, tmenvarant and tme-varyng, cntnuus and sampled-data, uncertan multple-nput sngleutput (MISO) and multple-nput multple-utput (MIMO) plants, and fr bth utput and nternal varable feedback. The QFT synthess technque fr hghly uncertan lnear tme-nvarant MIMO plants has the fllwng features: 1. The MIMO synthess prblem s cnverted nt a number f sngle-lp feedback prblems n whch parameter uncertanty, external dsturbances, and perfrmance tlerances are derved frm the rgnal MIMO prblem. The slutns t these snglelp prblems represent a slutn t the MIMO plant. 2. The desgn s tuned t the extent f the uncertanty and the perfrmance tlerances. Ths desgn technque s applcable t the fllwng prblem classes: 1. Sngle-nput sngle-utput (SISO) lnear-tme-nvarant (TI) systems 2. SISO nnlnear systems. 3. MIMO TI systems. 4. MIMO nnlnear systems.

3 38 Autmatc Flght Cntrl Systems atest evelpments 5. strbuted systems. 6. Sampled-data systems as well as cntnuus systems fr all f the precedng. Prblem classes 3 and 4 are cnverted nt equvalent sets f MISO systems t whch the QFT desgn technque s appled. The bjectve s t slve the MISO prblems,.e., t fnd cmpensatn functns whch guarantee that the perfrmance tlerances fr each MISO prblem are satsfed fr all P n. Ths chapter s essentally dvded nt tw parts. The frst part, cnsstng f Sectns 2 thrugh 4, presents the fundamentals f the QFT rbust cntrl system desgn technque fr the trackng and regulatr cntrl prblems. The secnd part cnssts f Sectn 5 whch fcuses n the applcatn f QFT technque t the flght cntrl desgn fr a certan Unmaned Aeral Vehcle (UAV). Ths s accmplshed by decmpsng the UAV s MIMO plant t 2 MISO plants whse cntrllers are bth synthszed usng QFT techque fr MISO systems. And the effectveness f bth cntrllers s verfed accrdng the dgtal smulatn results. Besdes, Sectns 6 thrugh 8 are abut summary f whle chapter, references and symbls used n the chapter. 2. Overvew f QFT 2.1 esgn bjectve f QFT Objectve f QFT s t desgn and mplement rbust cntrl fr a system wth structured parametrc uncertanty that satsfes the desred perfrmance specfcatns. 2.2 Perfrmance specfcatns fr cntrl system In many cntrl systems the utput y() t must le between specfed upper and lwer bunds, yt () U and y() t, respectvely, as shwn n Fg.1a. The cnventnal tme-dman fgures f mert, based upn a step nput sgnal rt () are shwn n Fg.1a. They are: M, peak P versht; t, rse tme; t r p, peak tme; and t s, settlng tme. Crrespndng system perfrmance specfcatns n the frequency dman are, B and B, the upper and lwer U bunds respectvely, peak versht m M, and the frequency bandwdth whch are m h shwn n Fg.1b. (a) tme dman respnse specfcatns (b) frequency dman respnse specfcatns Fg. 1. esred system perfrmance specfcatns

4 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 39 Assume that the cntrl system has neglgble sensr nse and suffcent cntrl effrt authrty, then fr a stable TI mnmum-phase plant, a TI cmpensatr may be desgned t acheve the desred cntrl system perfrmance specfcatns. 2.3 Implementatn f QFT desgn bjectve The QFT desgn bjectve s acheved by: epresentng the characterstcs f the plant and the desred system perfrmance specfcatns n the frequency dman. Usng these representatns t desgn a cmpensatr (cntrller). epresentng the nnlnear plant characterstcs by a set f TI transfer functns that cver the range f structured parametrc uncertanty. epresentng the system perfrmance specfcatns (see Fg.1) by TI transfer functns that frm the upper B and lwer B bundares fr the desgn. U educng the effect f parameter uncertanty by shapng the pen-lp frequency respnses s that the Bde plts f the J clsed-lp systems fall between the bundares B and B U, whle smultaneusly satsfyng all perfrmance specfcatns. Obtanng the stablty, trackng, dsturbance, and crss-cuplng (fr MIMO systems) bundares n the Nchls chart n rder t satsfy the perfrmance specfcatns. 2.4 QFT bascs Cnsder the cntrl system f Fg.2, where Gs () s a cmpensatr, Fs () s a preflter, and s the nnlnear plant wth structured parametrc uncertanty. T carry ut a QFT desgn: The nnlnear plant s descrbed by a set f J mnmum-phase TI plants,.e., { P( s)}( t 1,2,, J) whch defne the structured plant parameter uncertanty. t The magntude varatn due t the plant parameter uncertanty, ( j ), s depcted by P the Bde plts f the TI plants as shwn n Fg. 3 whch s fr a certan plant. J data pnts (lg magntude and phase angle), fr each value f frequency,, are pltted n the Nchls chart. A cntur s drawn thrugh the data pnts that descrbed the bundary f the regn that cntans all J pnts. Ths cntur s referred t as a template. It represents the regn f structured plant parametrc uncertanty n the Nchls chart and are btaned fr specfed values f frequency,, wthn the bandwdth (BW) f cncern. Sx data pnts (lg magntude and phase angle) fr each value f are btaned, as shwn n Fg. 4a, fr a certan example t plt the templates, fr each value f, as shwn n Fg. 4b. The system perfrmance specfcatns are represented by TI transfer functns, and ther crrespndng Bde plts are shwn n Fg. 3 by the upper and lwer bunds B U and B, respectvely. Fg. 2. Cmpensated nnlnear system

5 40 Autmatc Flght Cntrl Systems atest evelpments Fg. 3. TI plants (a) (b) (c) Fg. 4. (a) Bde plts f 6 TI plants; (b) template cnstructn fr =3 rad/sec; (c) cnstructn f the Nchls chart plant templates 2.5 QFT desgn The trackng desgn bjectve s t a. Synthesze a cmpensatr Gs () f Fg. 2 that results n satsfyng the desred perfrmance specfcatns f Fg. 1 results n the clsed-lp frequency respnses T shwn n Fg. 5 results n the ( j ) f Fg. 5 f the cmpensated system, beng equal t r smaller than ( j ) f Fg. 3 fr the uncmpensated system and that t s equal r less P than ( j ), fr each value f f nterest; that s: ( j ) ( j ) ( j ) P b. Synthesze a preflter Fs () f Fg. 2 that results n shftng and reshapng the T respnses n rder that they le wthn the B U and B bundares n Fg. 5 as shwn n Fg. 6.

6 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 41 Fg. 5. Clsed-lp respnses: TI plants wth G(s) Fg. 6. Clsed-lp respnses: TI plants wth G(s) and F(s) Therefre, the QFT rbust desgn technque assures that the desred perfrmance specfcatns are satsfed ver the prescrbed regn f structured plant parametrc uncertanty. 3. Insght t the QFT technque 3.1 Open-lp plant Cnsder a certan pstn cntrl system whse plant transfer functn s gven by K K a P() s t ss ( a) ss ( a) (1) where K K and 1,2,..., J. The lg magntude changes n a prescrbed range due t the a plant parameter uncertanty. The lp transmssn s () s defned as () s G() s P() s (2) t t 3.2 Clsed-lp frmulatn The cntrl rat T f the unty-feedback system f Fg. 2 s

7 42 Autmatc Flght Cntrl Systems atest evelpments T t Y t 1 (3) t The verall system cntrl rat T s gven by: T t Fs () () s t () s 1 ( s) t (4) 3.3 esults f applyng the QFT desgn technque The prper applcatn f the rbust QFT desgn technque requres the utlzatn f the prescrbed perfrmance specfcatns frm the nset f the desgn prcess, and the selectn f a nmnal plant P frm the J TI plants. Once the prper lp shapng f () s G() s P() s s accmplshed, a syntheszed Gs () s acheved that satsfes the desred perfrmance specfcatns. The last step f ths desgn prcess s the synthess f the preflter that ensures that the Bde plts f T all le between the upper and lwer bunds B and B. U 3.4 Benefts f QFT The benefts f the QFT technque may be summarzed as fllws: It results n a rbust desgn whch s nsenstve t structured plant parameter varatn. There can be ne rbust desgn fr the full, peratng envelpe. esgn lmtatns are apparent up frnt and durng the desgn prcess. The achevable perfrmance specfcatns can be determned n the early desgn stage. If necessary, ne can redesgn fr changes n the specfcatns quckly wth the ad f the QFT CA package. The structure f the cmpensatr (cntrller) s determned up frnt. There s less develpment tme fr a full envelpe desgn. 4. QFT desgn fr the MISO analg cntrl system 4.1 Intrductn The MIMO synthess prblem s cnverted nt a number f sngle-lp feedback prblems n whch parameter uncertanty, crss-cuplng effects, and system perfrmance tlerances are derved frm the rgnal MIMO prblem. The slutns t these sngle-lp prblems represent a slutn t the MIMO plant. It s nt necessary t cnsder the cmplete system characterstc equatn. The desgn s tuned t the extent f the uncertanty and the perfrmance tlerances. Here, we wll present an n-depth understandng and apprecatn f the pwer f the QFT technque thrugh apply QFT t a rbust sngle-lp MISO system, whch has tw nputs, a trackng and an external dsturbance nput, respectvely, and a sngle utput cntrl system.

8 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl The QFT methd (sngle-lp MISO system) Basc structure f a feedback cntrl system s gven n Fg.7, n whch represents the set f transfer functns whch descrbe the regn f plant parameter uncertanty, G s the cascade cmpensatr, and F s an nput preflter transfer functn. The utput yt () s requred t track the cmmand nput rt () and t reject the external dsturbances d () t and 1 d () t 2. The cmpensatr G n Fg. 7 s t be desgned s that the varatn f yt () t the uncertanty n the plant P s wthn allwable tlerances and the effects f the dsturbances d () t and d () t 1 2 n y() t are acceptably small. Als, the preflter prpertes f Fs () must be desgned t the desred trackng by the utput y() t f the nput rt (). Snce the cntrl system n Fg. 7 has tw measurable quanttes, rt () and y() t, t s referred t as a tw degree-f-freedm (OF) feedback structure. If the tw dsturbance nputs are measurable, then t represents a fur OF structure. The actual desgn s clsely related t the extent f the uncertanty and t the narrwness f the perfrmance tlerances. The uncertanty f the plant transfer functn s dented by the set and s llustrated as fllws. Gven that the plant transfer functn s { P} wheret 1,2,..., J (5) t K Ps () ss ( a) (6) where the value f K s n the range [1, 10] and a s n the range [-2, 2]. The desgn bjectve s t guarantee that T () s Y()/ s () s and T () s Y()/ s () s are members f the sets f acceptable and fr changes f K and a. In a feedback cntrl system, the prncpal challenge n the cntrl system desgn s t relate the system perfrmance specfcatns t the requrements n the lp transmssn functn s () GsPs () () n rder t acheve the desred benefts f feedback,.e., the desred reductn n senstvty t plant uncertanty and desred dsturbance attenuatn. The advantage f the frequency dman s that s () GsPs () () s smply the multplcatn f cmplex numbers. In the frequency dman t s pssble t evaluate j ( ) at every separately, and thus, at each, the ptmal bunds n j ( ) can be determned. Fg. 7. A feedback structure 4.3 QFT desgn prcedure The bjectve s t desgn the preflter Fs () and the cmpensatr Gs () f Fg.7 s that the specfed rbust desgn s acheved fr the gven regn f plant parameter uncertanty. The desgn prcedure t accmplsh ths bjectve s as fllws:

9 44 Autmatc Flght Cntrl Systems atest evelpments Step 1. Synthesze the desred trackng mdel. Step 2. Synthesze the desred dsturbance mdel. Step 3. Specfy the J TI plant mdels that defne the bundary f the regn f plant parameter uncertanty. Step 4. Obtan plant templates at specfed frequences that pctrally descrbe the regn f plant parameter uncertanty n the Nchls chart. Step 5. Select the nmnal plant transfer functn P() s. Step 6. etermne the stablty cntur ( U -cntur) n the Nchls chart. Step 7-9. etermne the dsturbance, trackng, and ptmal bunds n the Nchls chart. Step 10. Synthesze the nmnal lp transmssn functn () s G() s P() s that satsfes all the bunds and the stablty cntur. Step 11. Based upn Steps 1 thrugh 10, synthesze the preflter Fs (). Step 12. Smulate the system n rder t btan the tme respnse data fr each f the J plants. The fllwng sectns wll llustrate the desgn prcedure step by step. 4.4 Mnmum-phase system perfrmance specfcatns In rder t apply the QFT technque, t s necessary t synthesze the desred mdel cntrl rat based upn the system's desred perfrmance specfcatns n the tme dman. Fr the mnmum-phase TI MISO system f Fg. 7, the cntrl rats fr trackng and fr dsturbance rejectn are, respectvely, FsGsPs () () () Fss ()() T () s F() s T() s wth d () t d () t GsPs ( ) ( ) 1 s ( ) (7) Ps () P T wth r() t d () t GsPs ( ) ( ) 1 (8) 1 1 T wth r() t d () t GsPs ( ) ( ) 1 (9) Trackng mdels The QFT technque requres that the desred trackng cntrl rats be mdeled n the frequency dman t satsfy the requred gan K m and the desred tme dman perfrmance specfcatns fr a step nput. Thus, the system's trackng perfrmance specfcatns fr a smple secnd-rder system are based upn satsfyng sme r all f the step frcng functn fgures f mert (FOM) fr under-damped ( M, t, t, t, K ) and ver-damped p p s r m ( t, t, K ) s r m respnses, respectvely. These are graphcally depcted n Fg. 8. The tme respnses yt () U and y() t n ths fgure represent the upper and lwer bunds, respectvely, f the trackng perfrmance specfcatns; that s, an acceptable respnse yt () must le between these bunds. The Bde plts f the upper bund B U and lwer bund B fr m T ( j ) vs. are shwn n Fg. 9.

10 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 45 It s desrable t synthesze the cntrl rats crrespndng t the upper and lwer bunds T and T, respectvely, s that ( j ) ncreases as ncreases abve the 0-dB crssng U frequency (see Fg. 9b) f T cf U. Ths characterstc f ( j ) smplfes the prcess f syntheszng the lp transmssn () s G() s P() s as dscussed n Sec f ths chapter. T synthesze () s, t s necessary t determne the trackng bunds B ( j ) (see Sec. 4.9) whch are btaned based upn ( j ). Ths characterstc f ( j ) ensures that the trackng bunds B ( j ) decrease n magntude as ncreases. Fg. 8. System tme dman trackng perfrmance specfcatns (a) Ideal smple secnd-rder mdels (b) The augmented mdels Fg. 9. Bde plts f T An apprach t the mdelng prcess s t start wth a smple secnd-rder mdel f the desred cntrl rat T havng the frm U T U () s s 2 s ( s p )( s p ) 2 2 n n 2 2 n n 1 2 (10) 2 where p p and t T 4/ 4/ (the desred settlng tme). The cntrl rat n 1 2 s s n T s f Eq. (10) can be represented by an equvalent unty-feedback system s that U () where T U Ys () G () s eq () s (11) s () 1 G () s eq

11 46 Autmatc Flght Cntrl Systems atest evelpments 2 n G () s eq ss ( 2 ) n (12) The gan cnstant f ths equvalent Type1 transfer functn G () s s K eq 1 lm[ sg ( s)] eq s0 /2. n The smplest ver-damped mdel fr T () s s f the frm T Ys () K G () s eq () s s () ( s )( s ) 1 G () s 1 2 eq (13) where Geq 1 2 () s ss( ) 1 2 and K /( ). Selectn f the parameters 1 and s used t meet the 2 specfcatns fr ts and K 1. Once the deal mdels T ( j ) and T ( j ) are determned, the tme and frequency respnse U plts f Fgs. 8 and 9a, respectvely, can then be drawn. The hgh-frequency range n Fg. 9a s defned as, where b b s the mdel BW frequency f B U. In rder t acheve the desred characterstc f an ncreasng magntude f f B U fr, an cf ncreasng spread between B and B s requred n the hgh-frequency range (see Fg. 9b), U that s, B B (14) hf U must ncrease wth ncreasng frequency. Ths desred ncrease n s acheved by changng B and B U by augmentng T U wth a zer [see Eq. (15)] as clse t the rgn as pssble wthut sgnfcantly affectng the tme respnse. Ths addtnal zer rases the curve B fr the frequency range abve U cf. The spread can be further ncreased by augmentng T wth a negatve real ple [see Eq. (16)] whch s as clse t the rgn as pssble but far enugh away nt t sgnfcantly affect the tme respnse. Nte that the straght-lne Bde plt s shwn nly fr T. Ths addtnal ple lwers B fr ths frequency range. T U a s a a s z () s s s s s 2 2 ( / )( ) ( / )( ) n n ( )( ) n n 1 2 (15) T K K () s ( sa )( sa )( sa ) ( s )( s )( s ) (16) Thus, the magntude f ( j ) ncreases as, ncreases abve. cf

12 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 47 In rder t mnmze the teratn prcess n achevng acceptable mdels fr T () s and U T () s whch have an ncreasng ( j ), the fllwng prcedure may expedte the desgn prcess: (a) frst synthesze the secnd-rder mdel f Eq. (15) cntanng the zer at z a that meets the desred FOM; and (b) then, as a frst tral, select all three real 1 n ples f Eq. (16) t have the value f a a a. Fr succeedng trals, f 3 3 n 2 1 necessary, ne r mre f these ples are mved rght and/r left untl the desred specfcatns are satsfed. As llustrated by the slpes f the straght-lne Bde plts n Fg. 9b, selectng the value f all three ples n the range specfed abve nsures an ncreasng. Other pssbltes are as fllws: (c) the specfed values f t p and t s fr T may be such that a par f cmplex ples and a real ple need t be chsen fr the mdel respnse. Fr ths stuatn, the real ple must be mre dmnant than the cmplex ples, (d) dependng n the perfrmance specfcatns, T () s may requre tw real ples and a U zer "clse" t the rgn,.e., select z 1 very much less than p 1 and p 2 n rder t effectvely have an under-damped respnse. At hgh frequences (see Fg. 9b) must be larger than the actual varatn n the plant,. hf P Fr the case where y() t, crrespndng t T U, s t have an allwable large versht fllwed by a small tlerable undersht, a dmnant cmplex ple par s nt sutable fr T. An acceptable versht wth n undersht fr T can be acheved by T havng U U U tw real dmnant ples p 1 p 2, a dmnant real zer ( z 1 p 1 ) "clse"' t p 1, and a far ff ple p 3 p 2. The clseness f the zer dctates the value f M. Thus, a desgner selects a P ple-zer cmbnatn t yeld the frm f the desred tme-dman respnse sturbance rejectn mdels The smplest dsturbance cntrl rat mdel specfcatn s T ( j) Y( j / ( j)) a, a P cnstant, [the desred maxmum magntude f the utput based upn a unt-step dsturbance nput];.e., fr d 1 () t : y( t ) a, and fr: d p p 2 () t y() t a fr t t. Thus, the p x frequency dman dsturbance specfcatn s m T ( j) m a ver the desred specfed p BW (see Fg. 10). Fg. 10. Bde plts f dsturbance mdels fr T ( j )

13 48 Autmatc Flght Cntrl Systems atest evelpments 4.5 J TI plant mdels The smple plant f Eq. (17) Ka Ps () t ss ( a) (17) where K {1,10} and a {1,10}, s used t llustrate the MISO QFT desgn prcedure. The regn f plant parameter uncertanty may be descrbed by J TI plants, where 1,2,..., J whch le n ts bundary. 4.6 Plant templates f Ps (), Pj ( ) Wth GP, Eq. (7) yelds t m T m F m m F m T 1 The change n T due t the uncertanty n P, snce F s TI, s ( m T ) m T m F m 1 (18) (19) The prper desgn f and F, must restrct ths change n T s that the actual value f m T always les between B and B U f Fg. 9b. The frst step n syntheszng an s t make NC templates whch characterze the varatn f the plant uncertanty fr varus values f, ver a frequency range x h, where x cf. Fr the plant f Eq. (17) the values K = a = 1 represent the lwest pnt f each f the templates P( j ) and may be selected as the nmnal plant P fr all frequences. 4.7 Nmnal plant Whle any plant case can be chsen, t s a cmmn practce t select, whenever pssble, a plant whse NC pnt s always at the lwer left crner f the template fr all frequences fr whch the templates are btaned. 4.8 U-cntur (stablty bund) The specfcatns n system perfrmance n the tme dman (see Fg. 8) and n the frequency dman (see Fg. 9) dentfy a mnmum dampng rat fr the dmnant rts f the clsed-lp system whch crrespnds t a bund n the value f M M. On the p m NC ths bund n M M (see Fg. 11) establshes a regn whch must nt be penetrated p by the templates and the lp transmssn functns ( j ) t fr all. The bundary f ths regn s referred t as the stablty bund, the U-cntur, because ths becmes the dmnatng cnstrant n ( j ). Therefre, the tp prtn, ndcated by the crdnates efa, f the M cntur becmes part f the U-cntur. The frmatn f the U -cntur s dscussed n ths sectn. Fr the tw cases f dsturbance rejectn depcted n Fg. 7 the cntrl rats are, respectvely, as gven n Eqs. (8) and (9).

14 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 49 Fg. 11. U-cntur cnstructn (stablty cntur) Thus, t s necessary t synthesze an () s s that the dsturbances are prperly attenuated. Fr the present, nly ne aspect f ths dsturbance-respnse prblem s cnsdered, namely a cnstrant s placed n the dampng rat f the dmnant cmplex-ple par f T nearest the -axs. Ths dampng rat s related t the peak value f T( j ) j ( ) 1 ( j ) (20) Therefre, t s reasnable t add the requrement T 1 M (21) where M s a cnstant fr all and ver the whle range f parameter values. Ths results n a cnstrant n f the dmnant cmplex-ple par f T. Ths cnstrant can therefre be transfrmed nt a cnstrant n the maxmum value T max f Eq. (20). Ths results n lmtng the peak f the dsturbance respnse. A value f M can be selected t crrespnd t the maxmum value f T. Therefre, the tp prtn, efa as shwn n Fg.11, f the M-cntur n the NC, whch crrespnds t the value f the selected value f M, becmes part f the U-cntur. Fr a large class f prblems, as, the lmtng value f the plant transfer functn appraches K ' lm[ P( j )] where represents the excess f ples ver zers f Ps ().

15 50 Autmatc Flght Cntrl Systems atest evelpments 4.9 Trackng bunds B ( j ) Cnsder the plt f m P( j ) vs. P( j ) fr a plant shwn n Fg. 12 (the sld curve). Wth Gs () A 1and Fs () 1 n Fg. 7, P. The plt f m ( j ) vs. ( j ) s tangent t the M = 1dB curve wth a resnant frequency 1.1 m. If m M 2 m db s specfed fr m T, the gan A s ncreased, rasng m ( j ), untl t s tangent t the 2-dB M-curve. Fr ths example the curve s rased by m A 4.5 db( G A 1.679) and the resnant frequency s = m Nw cnsder that the plant uncertanty nvlves nly the varatn n gan A between the values f 1 and It s desred t fnd a cascade cmpensatr Gs (), n Fg. 7, such that the specfcatn 1 db m M 2 db s always mantaned fr ths plant gan varatn whle the m resnant frequency m remans cnstant. Ths requres that the lp transmssn ( j ) G( j) P( j) be syntheszed s that t s tangent t an M-cntur n the range f 1 db m M 2 db fr the entre range f 1 <A <1.679 and the resultant resnant frequency satsfes the requrement m m Fg. 12. g magntude-angle dagram It s assumed fr Eq. (19) that the cmpensatrs F and G are fxed (TI), that s, they have neglgble uncertanty. Thus, nly the uncertanty n P cntrbutes t the change n T gven by Eq. (19). The slutn requres that the actual mt ( j ) ( j ) db n Fg. 9b. Thus, t s necessary t determne the resultng cnstrant, r bund B ( j ), n ( j ). The prcedure s t select a nmnal plant P() s and t derve the bunds n the resultng nmnal lp transfer functn () s G() s P() s. As an llustratn, cnsder the plt f m P( j2) vs. P( j2) fr the plant f Eq. (17). As shwn n Fg. 13, the plant's regn f uncertanty P( j2) s gven by the cntur ABC,.e., m P( j2) les n r wthn the bundary f ths cntur. The nmnal plant transfer functn, wth K 1 and a 1, s

16 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 51 1 P() s ss ( 1) (22) and s represented n Fg. 13 by pnt A fr = 2 [-13.0 db, }. Nte, nce a nmnal plant s chsen, t must be used fr determnng all the bunds B ( j ). Fg. 13. ervatn f bunds B ( j ) n ( j ) fr = sturbance bunds B ( j ): CASE 1 Tw dsturbance nputs are shwn n Fg. 7. It s assumed that nly ne dsturbance nput exsts at a tme. Bth cases are analyzed. CASE 1 [ d () t u (), t d () t ] CONTO ATIO. Frm Fg. 7, the dsturbance cntrl rat fr nput d () t s 2 Substtutng 1 / nt Eq. (23) yelds 1 T () s (23) 1 T () s 1 (24) ths equatn has the mathematcal frmat requred t use the NC. Over the specfed BW t s desred that T ( j ) 1, whch results n the requrement, frm Eq.(24), that ( j ) 1,.e.,

17 52 Autmatc Flght Cntrl Systems atest evelpments T ( j ) 1 ( j ) ( j ) ISTUBANCE ESPONSE CHAACTEISTIC. A tme-dman trackng respnse characterstc based upn rt () u () t ften specfes a maxmum allwable peak 1 versht M. In the frequency dman ths specfcatn may be apprxmated by p Y( j ) M ( j) T ( j) M M m P (25) ( j ) The crrespndng tme- and frequency-dman respnse characterstcs, based upn the step dsturbance frcng functn d 2 () t u 1 () t, are, respectvely, Yt () M () t fr t t P x (26) dt () and Y( j ) M ( j ) T ( j) (27) m P j ( ) 4.11 sturbance bunds B ( j ): CASE 2 CASE 2 [ d () t u (), t d () t 0] CONTO ATIO. Frm Fg. 7, the dsturbance cntrl rat fr the nput d () t s 1 P( j ) T ( j ) (28) 1 G( j ) P( j ) Assumng pnt A f the template represents the nmnal plant P. Eq. (28) s multpled by P / P and rearranged as fllws: T P 1 P P P 1 P P P W G GP P P P (29) where W ( P / P) (30) Thus Eq.(29) wth m T yelds m W m P (31)

18 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 53 ISTUBANCE ESPONSE CHAACTEISTICS. Based n Eq. (25), the tme and frequency-dman respnse characterstcs, fr a unt-step dsturbance frcng functn, are gven, respectvely, by and where t s the peak tme. p yt ( ) p M () t y( t ) (32) p p dt () Y( j ) M ( j ) T ( j) (33) m p j ( ) 4.12 The cmpste bundary B ( j ) The cmpste bund B ( j ) that s used t synthesze the desred lp transmssn transfer functn () s s btaned n the manner shwn n Fg. 14. The cmpste bund B ( j ), fr each value f, s cmpsed f thse prtns f each respectve bund B ( j ) and B ( j ) that are the mst restrctve. Fr the case shwn n Fg. 14a the bund B ( j ) s cmpsed f thse prtns f each respectve bund B ( j ) and B ( j ) that have the largest values. Fr the stuatn f Fg. 14b, the utermst f the tw bundares B ( j ) and B ( j ) becmes the permeter f B ( j ). The stuatns f Fg. 14 ccur when the tw bunds have ne r mre ntersectns. If there are n ntersectns, then the bund wth the largest value r wth the utermst bundary dmnates. The syntheszed ( j ), fr the stuatn f Fg. 14a, must be n r just abve the bund B ( j ). Fr the stuatn f Fg. 14b the syntheszed ( j ) must nt le n the nterr f the B ( j ) cntur. Fg. 14. Cmpste B ( j ) 4.13 Shapng f ( j ) A realstc defntn f ptmum n an TI system s the mnmzatn f the hgh-frequency lp gan K whle satsfyng the perfrmance bunds. Ths gan affects the hgh-frequency respnse snce lm[ j ( )] Kj ( ) where s the excess f ples ver zers assgned

19 54 Autmatc Flght Cntrl Systems atest evelpments t ( j ). Thus, nly the gan K has a sgnfcant effect n the hgh-frequency respnse, and the effect f the ther parameter uncertanty s neglgble. Als, the mprtance f mnmzng the hgh-frequency lp gan s t mnmze the effect f sensr nse whse spectrum, n general, les n the hgh-frequency range. Fr the plant f Eq. (17), the shapng f ( j ) s shwn by the dashed curve n Fg. 15. A pnt such as m ( j 2) must be n r abve the curve labeled B ( j 2). Further, n rder t satsfy the specfcatns, ( j ) cannt vlate the U-cntur. In ths example a reasnable ( j ) clsely fllws the U-cntur up t 40 rad/sec and stays belw t abve 40 as shwn n Fg 15. Addtnal specfcatns are = 4,.e., there are 4 ples n excess f zers, and that t als must be Type 1 (ne ple at the rgn).a representatve prcedure fr chsng a ratnal functn () s whch satsfes the abve specfcatns s nw descrbed. It nvlves buldng up the functn w ( j ) ( j) P( j) [ K G ( j)] (34) k k k k0 where fr k = 0, G 10, and K w k0 K k In rder t mnmze the rder f the cmpensatr, a gd startng pnt fr "buldng up" the lp transmssn functn s t ntally assume that ( j ) = P ( j ) as ndcated n Eq. 0 (34). ( j ) s bult up term-by-term n rder t stay just utsde the U-cntur n the NC f Fg. 15. The frst step s t fnd the B ( j ) whch dmnates ( j ). Fg. 15. Shapng f ( j ) n the Nchls chart fr the plant f Eq. (17)

20 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl esgn f the preflter Fs () esgn f a prper () s guarantees nly that the varatn n T ( j ),.e., T, s less than r equal t that allwed. The purpse f the preflter s t pstn m T( j ) wthn the frequency dman specfcatns. Fr the example f ths chapter the magntude f the frequency respnse must be wthn the bunds B U and B shwn n Fg. 9b, whch are redrawn n Fg. 16. A methd fr determnng the bunds n Fs () s as fllws: Place the nmnal pnt A f the plant template n the ( j ) pnt f the ( j ) curve n the NC (see Fg. 17). Traversng the template, determne the maxmum m T and max mnmum m T, values f mn j ( ) m T( j ) (35) 1 ( j ) btaned frm the M-cnturs. These values are pltted as shwn n Fg. 16. The trackng cntrl rat s T F/[1 ] and m T ( j ) m F( j ) m T( j ) (36) The varatns n Eqs. (35) and (36) are bth due t the varatn n P; thus ( j ) m T m T B B (37) max mn U Fg. 16. equrements n Fs () Fg. 17. Preflter determnatn

21 56 Autmatc Flght Cntrl Systems atest evelpments If values f ( j ), fr each value, le exactly n the trackng bunds B ( j ), then. Therefre, based upn Eq. (36), t s necessary t determne the range n db by whch m T( j ) must be rased r lwered t ft wthn the bunds f the specfcatns by use f the preflter F( j ). The prcess s repeated fr each frequency crrespndng t the templates used n the desgn f ( j ). Therefre, n Fg. 18 the dfference between the m T m T and the m T m T curves yelds the requrement fr m F( j ),.e., frm U max mn Eq. (36). m F( j ) m T ( j) m T( j) (38) Fg. 18. Frequency bunds n the preflter Fs () The prcedure fr desgnng Fs () s summarzed as fllws: 1. Use templates n cnjunctn wth the ( j ) plt n the NC t determne T and T max mn fr each. Ths s dne by placng P( j ) wth ts nmnal pnt n the pnt m ( j ). Then use the M-cnturs t determne T max ( j ) and T mn ( j ) (see Fg. 17). 2. Obtan the values f m T and m T fr varus values f a, frm Fg. 9b. U 3. Frm the values btaned n steps 1 and 2, plt m T m T U max and m T m T mn vs. as shwn n Fg Use straght-lne apprxmatns t synthesze an Fs () s that m F( j ) les wthn the plts f step 3. Fr step frcng functns the resultng Fs () must satsfy lm[ Fs ( )] 1 (39) s Basc desgn prcedure fr a MISO system The basc cncepts f the QFT technque are explaned by means f a desgn example. The system cnfguratn shwn n Fg. 7 cntans three nputs. The frst bjectves are t track a step nput rt () u () t wth n steady-state errr and t satsfy the perfrmance 1 specfcatns f Fg. 8. An addtnal bjectve s t attenuate the system respnse caused by

22 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 57 external step dsturbance nputs d () t and d () t. An utlne f the basc desgn prcedure fr 1 2 the QFT technque, as appled t a mnmum-phase plant, s as fllws: 1. Synthesze the trackng mdel cntrl rat T () s n the way descrbed n Sec. 4.4, based upn the desred trackng specfcatns (see Fgs. 8 and 9b). 2. Synthesze the dsturbance-rejectn mdel cntrl rats T () s n the manner descrbed n Sec based upn the dsturbance-rejectn specfcatns. 3. Obtan templates f P( j ) that pctrally descrbe the plant uncertanty n the Nchls chart fr the desred pass-band frequency range. 4. Select a nmnal plant frm the set f Eq. (5) and dente t as P() s. 5. etermne the U-cntur based upn the specfed values f ( j ) fr trackng, M fr dsturbance rejectn, and V fr the unversal hgh frequency bundary (UHFB) n cnjunctn wth steps 6 thrugh Use the data f steps 2 and 3 and the values f ( j ) (see Fg. 10) t determne the dsturbance bund B ( j ) n the lp transmssn ( j ) G( j ) P ( j ). Fr mnmumphase systems ths requres that the syntheszed lp transmssn m ( j ) must be n r abve the curve fr m B ( j ) n the Nchls dagram (see Fg. 15 assumng B B ). 7. etermne the trackng bund B ( j ) n the nmnal transmssn ( j ) G( j ) P ( j ), usng the trackng mdel (step 1), the templates P( j ) (step 3), the values f ( j ) (see Fg. 9b), and M [see Eq.(21)]. Fr mnmum-phase systems ths requres that the syntheszed lp transmssn satsfy the requrement that m ( j ) s n r abve the curve fr m B ( j ) n the Nchls dagram. 8. Plt curves f m B ( j ) versus B ( j ) and m B ( j ) versus B ( j ) n the d same NC. Fr a gven value f at varus values f the angle, select the value f m B ( j ) r m B ( j ), whchever s the largest value (termed the "wrst" r "mst severe" bundary). raw a curve thrugh these pnts. The resultng plt defnes the verall bundary m B ( j ) vs.. epeat ths prcedure fr suffcent values f. 9. esgn ( j ) t be as clse as pssble t the bundary value B ( j ) by selectng an apprprate cmpensatr transfer functn G( j ). Synthesze an ( j ) G( j) P ( j) usng the m B ( j ) bundares and U-cntur s that m ( j ) s n r abve the curve fr m B ( j ) n the Nchls dagram. 10. Based upn the nfrmatn avalable frm steps 1 and 9, synthesze an Fs () thse results n a m T [Eq. (7)] vs. that les between B and B f Fg. 9b. U 11. Obtan the tme-respnse data fr y() t : (a) wth dt () u () t and rt () 0 and (b) wth 1 rt () u () t and dt () 0 fr suffcent pnts arund the parameter space descrbng the 1 plant uncertanty. 5. bust QFT flght cntrl desgn fr a certan UAV 5.1 Intrductn Unmanned Aeral Vehcles (hereafter referred as UAVs) play a very mprtant rle n mdern war. Whereas flght stablty f UAVs s easly affected by arflw, mdel perturbatn and ther uncertanty. T enhance flght stablty and rbustness f UAVs,

23 58 Autmatc Flght Cntrl Systems atest evelpments H cntrl, QFT technque, lnear quadratc Gaussan (QG) have been appled t UAVs flght cntrl system at present. Cmparatvely, QFT can take uncertanty s scpes and perfrmance requrements nt accunt, analyze and desgn rbust cntrller n Nchls chart quanttatvely n rder t make the pen-lp frequency curve cmply wth bundary cndtns and have rbust stablty and perfrmance rbustness. QFT has been wdely used n aerspace feld and s mature fr rbust cntrller desgn f TI/SISO system. But QFT desgn fr MIMO system stll faces many dffcultes. In vew f the characterstcs f a certan small UAV whch used n trackng and survellance, a nvel QFT cntrller desgn methd fr the UAV s lateral mtn s ntrduced n ths sectn. 5.2 QFT desgn fr MIMO systems Overvew The QFT desgn fr MIMO systems s based upn the mathematcal means whch results n 2 the representatn f a MIMO cntrl system by m MISO equvalent cntrl systems. The hghly structured uncertan TT MIMO plant has the fllwng features: 1. The synthess prblem s cnverted nt a number f sngle-lp prblems, n whch structured parameter uncertanty, external dsturbance, and perfrmance tlerances are derved frm the rgnal MIMO prblem. The slutns t these sngle-lp prblems are guaranteed t wrk fr the MIMO plant. It s nt necessary t cnsder the system characterstc equatn. 2. The desgn s tuned t the extent f the uncertanty and the perfrmance tlerances. The desgn fr a MIMO system, as stated prevusly, nvlves the desgn f an equvalent set f MISO system feedback lps. The desgn prcess fr these ndvdual lps s the same as the desgn f a MISO system descrbed n prevus sectns. Pure mathematcal transfrmatn methd used n QFT desgn fr MIMO systems tends t cause a larger super-margn desgn and s very cmplcated when system s f hgher rder. Cmparatvely, Bascally Nn-nteractng (hereafter referred as BNIA) s cmmnly used n practcal applcatns. Nte that prncple f BNIA, whch wll be neglgble here, can be fund n references f ths chapter Nn-nteractng (BNIA) lps A BNIA lp s ne n whch the utput y () s due t the nput r( s) s deally zer. Plant K j uncertanty and lp nteractn (crss-cuplng) makes the deal respnse unachevable. Thus, the system perfrmance specfcatns descrbe a range f acceptable respnses fr the cmmanded utput and a maxmum tlerable respnse fr the uncmmanded utputs. The uncmmanded utputs are treated as crss-cuplng effects. Fr an TI plant havng n parameter uncertanty, t s pssble t essentally acheve zer crss-cuplng effects,.e., the utput y 0. Ths desred result can be acheved by pst K multplyng P by a matrx W t yeld:

24 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 59 P PW [ p ] where p 0 fr j n jn jn resultng n a dagnal P n matrx fr P representng the nmnal plant case n the set. Wth plant uncertanty the ff-dagnal terms f P n wll nt be zer but "very small" n cmparsn t P, fr the nnnmnal plant cases n. In sme desgn prblems t may be necessary r desred t determne a P n upn whch the QFT desgn s accmplshed. ng ths mnmzes the effrt requred t acheve the desred BW and mnmzes the crsscuplng effects. 5.3 QFT desgn and smulatn fr a certan UAV s lateral mtn QFT apprach fr MIMO system wll be appled t a certan UAV s lateral mtn n ths sectn Mathematcal mdel f the UAV State equatn f the UAV s generally expressed as: xt () Axt () But () yt Cxt () (40) where X p T ; u T ; Y p T ; s sdeslp angle, p s rll a r rc ac angle rate, s yaw angle rate, s rll angle, a s alern deflectn angle, rc s rudder deflectn angle, ac s rudder deflectn angle cmmand nput, A, BCare, system matrx, nput matrx and nput-utput matrx respectvely. By way f wnd tunnel test and mathematc methd, matrces A, B and C n eqs.(40) fr the small UAV can be derved System decmpstn The UAV s lateral state equatn descrbed n Eq.(40) has tw nputs and fur utputs. Accrdng t QFT apprach fr MIMO system, we decmpse Eq.(40) nt tw MISO subsystems usng BNIA, ne s yaw lp (lp I) subsystem, the ther s rll lp (lp II) subsystem. QFT cntrl structures f bth lps are gven n Fg.19 and Fg.20. Fg. 19. QFT cntrl structure f lp I Fg. 20. QFT cntrl structure f lp II

25 60 Autmatc Flght Cntrl Systems atest evelpments where, are sdeslp angle nput and rll angle nput respectvely; g c c 1, g 2 are QFT cntrllers; f, f are QFT preflters; c , c 22 are dsturbance nputs; q 11, q 22 s cntrlled plants. ecmpsed state equatn has relatnshp wth that f the rgnal system as fllws: A A, B B, C c c c C Transfer functn matrces P f decmpsed plant can be easly derved as p PC sia B ( ) c c c p21 p22 where p 11 s the transfer functn frm rc t ; p represents the transfer functn frm t 22 ac ; p s the transfer functn frm 12 rc t, p represents the transfer functn frm t. 21 ac Next, we adpt 5 flght states t develp the QFT cntrllers f bth lps QFT desgn fr lp I Fr lp I, we ensure g 1 ( s) and f 11 ( s) meet requrements f rbust stablty when c acts as cmmand nput and c 11 as dsturbance nput. Besdes, bth subsystems shuld wn deal trackng perfrmance and preferable nse restrant capablty. 1. Selectn f Perfrmance Indces. Trackng perfrmances ndces f sdeslp angle are versht % 2%, settlng tme t 6%. Gven the rgnal mdel f upper trackng bundary s s p T U ( j ) s 2 n s n n (41) Accrdng t % and t s, dampng rat and natural scllatn frequency s adpted n as 0.78 and Add a zer (z=-1) as clse t the rgn as pssble wthut sgnfcantly affectng the tme respnse(see Sec.4.4.1). Ths addtnal zer rases trackng bundary curve abve, the fnal transfer functn f trackng curve s upper bundary s cf T U ( j ) s ( s 1) 1.4s0.806 (42) the lwer bundary rgnal mdel f trackng curve as T 0.9 ( j ) (43) s 0.9 Addng tw ples (p1=-1, p2=-4), whch lcate n left half s-plane t ensure stablty f and are as clse t the rgn as pssble but far enugh away nt t sgnfcantly affect T the tme respnse (see Sec.4.4.1), t eq. (43) t make lwer trackng bundary separate frm

26 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 61 upper trackng bundary when upper trackng bundary crss ver 0 db lne, then the fnal lwer bundary transfer functn s T 3.6 ( j ) ( s0.9) s1 s4 (44) Stablty perfrmance ndex and rbust perfrmance ndex are respectvely q s g s q s g s and q s q s g s 0.1 Crrespndng mnmum ampltude margn and phase margn are respectvely and K m =5.5155dB u , cs (0.5 / 1) m 2. Plant Template and Brder Calculatn fr p I. Accrdng t the requrements f perfrmance ndex, generate the trackng respnse bundary, rbust stablty bundary and nference rejectn bundary n Nchls chart. 3. Cntrller and Preflter esgn fr p I. In Fg. 21(a), the pen-lp frequency characterstcs curve (nted by black sld lne) f the nmnal plant (crrespndng t G(s) =1) and the cmpund bundary (the regn embraced by green and red sld lne) are drawn up n Nchls chart. Apparently, the pen-lp frequency curve lcates under trackng perfrmance bundary curve, pen-lp frequency characterstcs curve crss ver the nstablty bundary (red sld rng lne n Fg. 21(a)) whch make the MISO system f lp I nstable r unsatsfactry fr crrespndng perfrmance requrements. S, t s necessary t enlarge the cntrller gan and ntrduce nt dynamc cmpensatn element t shape the pen lp frequency characterstc curve t ensure shaped pen-lp frequency characterstc meet the requrtments f stablty and dynamc perfrmance ndcs. Usng MATAB QFT tlbx, we get g 1 s s s s/ s/ / / (45) f 11 s s /0.6 1 (46) The pen-lp frequency characterstcs curve wth G(s) s shwn n Fg.21 (b). Clearly, the shaped curve des nt crss ver the nstablty regn (red sld rng lne),.e. the shaped system s stable. Besdes, the characterstc f trackng bundary s met.

27 62 Autmatc Flght Cntrl Systems atest evelpments (a) Open-lp frequency respnse when G(s) =1 (b) Open-lp frequency respnse wth cntrller Fg. 21. Open lp frequency characterstcs n Nchls Chart 4. Verfcatn and Smulatn fr p I. Clsed-lp system stablty margn analyss curve, nference rejectn bundary analyss curves and trackng bundary analyss curves n lp I are gven n Fg.22,Fg.23 and Fg.24. Clearly, the stablty margn curve, nference rejectn bundary curve and trackng bundary curve are all under the stablty perfrmance ndex curve, the perfrmance ndex curve and between the upper and lwer bundares f trackng curves. Obvusly, Clsed-lp cntrl system satsfes the perfrmance requrements n lp I Magntude /db Stablty perfrmance ndex curve Stablty margn curve Fg. 22. Stablty margn Frequency /rad/sec

28 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl bust perfrmance ndex curve Interfence rejectn curve Magntude /db Frequency /rad/sec Fg. 23. sturbance rejectn bundary 0-10 Magntude /db Upper bundary wer bundary Clsed-lp utput1 Clsed-lp utput1 Fg. 24. Trackng bundary Frequency /rad/sec The tme-dman smulatn results f clsed-lp system under 5 desgn envelpes are shwn n Fg.25 and Fg.26. The unt step-respnse f sdeslp angle les between the upper and lwer bundary respnse curve; the unt step-respnse f dsturbance nput are lcated under the gven bundary. Apparently, the clsed-lp system satsfes the requrements f rbust stablty and trackng bundary requrements, and wns strng dsturbance rejectn capablty.

29 64 Autmatc Flght Cntrl Systems atest evelpments Sdeslp angle/degree Upper bundary 0.4 wer bundary flght state 1 flght state flght state 3 flght state 4 flght state Tme/secnd Fg. 25. The unt step respnse f Sdeslp angle/degree flght state 1 flght state flght state 3 flght state 4 flght state Tme/secnd Fg. 26. The unt step respnse f wth dsturbance QFT desgn fr lp II QFT desgn fr lp II s smlar t that fr lp I. 1. Selectn f Perfrmance Indces. Trackng perfrmance ndces f rll angle s versht % 5% and settlng tme 12s, the upper and lwer bundary trackng curve are respectvely t s T U 0.25(1.7s 1) ( j) 2 s 0.78s0.25 (47)

30 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 65 T 3.6 ( j ) ( s0.9) s1 s4 (48) Stablty perfrmance ndex and rbust perfrmance ndex are defned as q () s g s 1 q ( s) g s 1.1 and q () s 22 1 q ( s) g s Mnmum ampltude margn and phase margn are B and respectvely. 2. Cntrller and Preflter esgn fr p II. Smlar t lp I, usng MATAB QFT tlbx, we can get 2 s s s s g () s (49) f 22 () s s (50) 3. Verfcatn and Smulatn fr lp II. Clsed-lp system satsfes requrements f rbust stablty and trackng bundary requrements and wns strng dsturbance rejectn capablty Perfrmance analyss f QFT cntrller fr the UAV s lateral mtn QFT cntrl structure fr the UAV s lateral mtn s shwn n Fg.27.Gven and c c are 0, the ntal value f s 5, the ntal sdeslp angle s 1, substtute the UAV s lateral state equatn, g (), s f (), s g (), s f () s, mdels f rudder and alerns nt Fg.27. The smulatn results are shwn n Fg.28 and Fg.29. The versht f s abut and settlng tme s abut 1 secnd. The settlng tme f yaw angle rate, rll angle rate and rll angle are all abut 0.1 secnd. Besdes, the ntal value f sdeslp angle almst have n nfluence n rll angle respnse, the settlng tme f yaw angle rate, rll angle rate s n mre than 1 secnd. Clearly, QFT cntrller fr the UAV s lateral mtn satsfes the requrements f perfrmance ndces, wn better flght stablty and rbustness. Fg. 27. QFT cntrl structure fr the UAV s lateral mtn

31 66 Autmatc Flght Cntrl Systems atest evelpments Sdeslp angle/degree flght state 1 flght state 2 flght state 3 flght state 4 flght state 5 ll angle/degree flght state 1 flght state 2 flght state 3 flght state 4 flght state Tme/secnd (a) Sdeslp angle Tme/secnd (b) ll angle Yaw angle rate/degree/secnd flght state 1 flght state 2 flght state 3 flght state 4 flght state Tme/secnd (c) Yaw angle rate ll angle rate/degre/secnd flght state 1 flght state 2 flght state 3 flght state 4 flght state Tme/secnd (d) ll angle rate Fg. 28. espnses f sdeslp angle, rll angle, yaw angle rate and rll angle rate when 0 5 ll angle/degree flght state 1 flght state 2 flght state 3 flght state 4 flght state Tme/secnd (a) Sdeslp angle ll angle/degree flght state 1 flght state 2 flght state 3-1 flght state 4 flght state tme/secnd (b) ll angle

32 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 67 Yaw angle rate/degree/secnd flght state 1 flght state 2 flght state 3 flght state 4 flght state Tme/secnd ll angle rate/degree/secnd x 10-3 flght state 1-15 flght state 2 flght state 3 flght state 4 flght state Tme/secnd (c) Yaw angle rate (d) ll angle rate Fg. 29. espnses f sdeslp angle, rll angle, yaw angle rate and rll angle rate when Summary Ths chapter s devted t presentng an vervew and n-depth expressn f QFT n rder t enhance the understandng and apprecatn f the pwer f the QFT technque. Then, A QFT desgn f rbust cntrller fr a certan UAV s lateral mtn, whch s a MIMO system, s prpsed base n BNIA prncple n rder t shw hw t apply QFT n flght cntrl system f UAVs. Meantme, the smulatn results shw that the QFT cntrller wn better rbust stablty and superr dynamc characterstcs whch verfy the valdty f presented methd. 7. Symbls & termnlgy T Acceptable cmmand r trackng nput-utput respnses A set f T T Acceptable dsturbance nput-utput respnses A set f T P MISO plant wth uncertanty A set f P MIMO Multple-nput multple-utput; mre than ne trackng and/r external dsturbance nputs and mre than ne utput MISO Multple-nput sngle-utput; a system havng ne trackng nput, ne r mre external dsturbance nputs, and a sngle utput B ( jw ), B ( jw ), B ( jw ) The dsturbance, trackng, and ptmal bunds n ( j ) fr the MISO K O system The frequency bandwdth h ( j ) The magntude varatn due t the plant parameter uncertanty P

33 68 Autmatc Flght Cntrl Systems atest evelpments m g magntude TI near-tme-nvarant FOM fgure f mert The symbl fr bandwdth frequency f the mdels b m The resnant frequency, th Phase margn frequency fr a MISO system and fr the lp f a MIMO system, respectvely Samplng frequency s, { r } The trackng nput fr a MISO system and the trackng nput vectr fr a MIMO system, respectvely B U mt The m f the desred trackng cntrl rat fr the upper bund f the MISO system B U mt The m f the desred trackng cntrl rat fr the lwer bund f the MISO system B Stablty bunds fr the dscrete desgn s ( j ) The (upper) value f m T ( j ) fr MISO system ( j ) The db dfference between the augmented bunds f B U and B n the hgh hf frequency range fr a MISO system ( j ) The db dfference between B and B fr a gven frequency, fr a MISO system U FF, { f j } The preflter fr a MISO system and the mxm preflter matrx fr a MIMO system respectvely GG, { f j } The cmpensatr r cntrller fr a MISO system and the mxm cmpensatr r cntrller matrx fr a MIMO system, respectvely. Fr a dagnal matrx G { f j }, The phase margn angle fr the MISO system and fr the th lp f the MIMO system, respectvely J The number f plant transfer functns fr a MISO system r plant matrx fr a MIMO system that descrbes the regn f plant parameter uncertanty where = 1, 2...J dentes the partcular plant case n the regn f plant parameter uncertanty The excess f ples ver zers f a transfer functn, The ptmal lp transmssn functn fr the MISO system and the th lp f the MIMO system, respectvely M, M The specfed clsed-lp frequency dman vershts cnstrant fr the MISO system and fr the th lp f a MIMO system, respectvely. Ths versht cnstrant may be dctated by the phase margn angle fr the specfed lp transmssn functn P( j ) Scrpt cap tee n cnjunctn wth P dentes a template,.e., P( j ) and Q( j ) frequency, fr a MISO and MIMO plants respectvely T The desred MISO trackng cntrl rat that satsfes the specfed upper bund U FOM

34 Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl 69 T The desred MISO trackng cntrl rat that satsfes the specfed lwer bund FOM T The desred MISO dsturbance cntrl rat whch satsfes the specfed FOM UAV Unmanned Aeral Vehcle BNIA Bascally Nn-nteractng 8. Acknwledgement The wrk f ths chapter s supprted by Natural Scence Basc esearch Plan n Shaanx Prvnce f Chna (Prgram N. 2011GQ8005) and Nrthwestern Plytechncal Unversty Fundatn fr Fundamental esearch (N. :NPU-FF-JC ) 9. eferences Chen Huamn: An Integrated QFT/EA Cntrller esgn Methd fr a UAV s ateral Flght Cntrl System, Mechancal Scence and Technlgy, Vl.27-3(2008),p Cnstantne H. Hups,Steven J. asmussen. Quanttatve Feedback Thery: Fundamentals and Applcatns[M], Marcel ekker, Inc. New Yrk, Basel Hrwtz I. M, and M. Sd, "Synthess f Feedback Systems wth arge Plant Ignrance fr Prescrbed Tme man Tlerances," Int. J. f Cntrl, vl. 16, pp , Hrwtz, I. M. and C. echer, "esgn 3x3 Multvarable Feedback System wth arge Plant Uncertanty," ht. J. Cntrl, vl. 33, pp ,1981. Ibd, "Synthess f Feedback Systems wth Nn-near Tme Uncertan Plants t Satsfy Quanttatve Perfrmance Specfcatns," IEEE Prc., vl. 64, pp ,1976. Hups, C. H. "Quanttatve Feedback Thery (QFT) fr the Engneer: A Paradgm fr the esgn f Cntrl Systems fr Uncertan Systems," W-T , AF Wrght abratry, Wrght-Pattersn AFB, OH, 1995 (Avalable frm Natnal Techncal Infrmatn Servce, 5285 Prt yal ad, Sprngfeld, VA 22151, dcument number A-A ) Hups, C. H. and P.. Chandler, Edtrs: "Quanttatve Feedback Thery Sympsum Prceedngs," W-T , Wrght abratres, Wrght-Pattersn AFB,OH, O Yanv,Y Chat:A Smplfed Mult-Input Mult-Output Frmulatn fr Quanttatve Feedback Thery, Jurnal f ynamc Systems, Measurement, and Cntrl,Vl.114-6(1998),p.179. eynlds, O.., M Pachter, and C. H. Hups. "esgn f a Subsnc Flght Cntrl System fr the Vsta F-16 Usng Quanttatve Feedback Thery, "Prceedngs f the Amercan Cntrl Cnference, pp ,1994. Thmpsn,. F., and O.. I. Nwkah, "Optmal p Synthess n Quanttatve Feedback Thery," Prceedngs, f the Amercan Cntrl Cnference, San eg, CA, pp ,

35 70 Autmatc Flght Cntrl Systems atest evelpments Trsen,. W., M, Pachter, and C. H. Hups, "Frmatn Flght Cntrl Autmatn," Prceedngs f the Amercan Insttute f Aernautcs and Astrnautcs (AIAA) Cnference, pp , Scttsdale, AZ,

36 Autmatc Flght Cntrl Systems - atest evelpments Edted by r. Thmas mbaerts ISBN Hard cver, 204 pages Publsher InTech Publshed nlne 18, January, 2012 Publshed n prnt edtn January, 2012 The hstry f flght cntrl s nseparably lnked t the hstry f avatn tself. Snce the early days, the cncept f autmatc flght cntrl systems has evlved frm mechancal cntrl systems t hghly advanced autmatc fly-by-wre flght cntrl systems whch can be fund nwadays n mltary jets and cvl arlners. Even tday, many research effrts are made fr the further develpment f these flght cntrl systems n varus aspects. ecent new develpments n ths feld fcus n a wealth f dfferent aspects. Ths bk fcuses n a selectn f key research areas, such as nertal navgatn, cntrl f unmanned arcraft and helcpters, trajectry cntrl f an unmanned space re-entry vehcle, aerservelastc cntrl, adaptve flght cntrl, and fault tlerant flght cntrl. Ths bk cnssts f tw majr sectns. The frst sectn fcuses n a lterature revew and sme recent theretcal develpments n flght cntrl systems. The secnd sectn dscusses sme cncepts f adaptve and fault-tlerant flght cntrl systems. Each technque dscussed n ths bk s llustrated by a relevant example. Hw t reference In rder t crrectly reference ths schlarly wrk, feel free t cpy and paste the fllwng: Xajun Xng and ngl Yuan (2012). Quanttatve Feedback Thery and Its Applcatn n UAV s Flght Cntrl, Autmatc Flght Cntrl Systems - atest evelpments, r. Thmas mbaerts (Ed.), ISBN: , InTech, Avalable frm: InTech Eurpe Unversty Campus STeP Slavka Krautzeka 83/A jeka, Crata Phne: +385 (51) Fax: +385 (51) InTech Chna Unt 405, Offce Blck, Htel Equatral Shangha N.65, Yan An ad (West), Shangha, , Chna Phne: Fax: